Question

HW3: Problem 7 Previous Problem Problem List Next Problem (1 point) Fundamental Existence Theorem for Linear Differential Equations Given the IVP dz1 d"y d" - 4.(2) +4-1(2) +...+41 () dy +40()y=g(2) dr y(t) = yo, y(t)= y yn-1 (3.) = Yn1 If the coefficients (1),..., Go() and the right hand side of the equation g(1) are continuous on an interval I and if (1) #0 on I then the IVP has a unique solution for the point I, EI that exists on the whole interval I. d It is useful to introduce an operator notation for derivatives. In particular we set D= which allows us to write the differential equation above as. dr (2.(z)D) ++1E)D(+1) +...+41(z)D + 20(z)) y=g() A note about putting in your answers. Some problems have may have answer blanks that require you to enter an intervals. Intervals can be written using interval notation: (2, 3) is the numbers I with 2<I<3 while (2, 3) is the interval 2 <I<3 If you want oo type inf, for example for the interval (-00,00) you would type :

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Answer 1

solution: (1) for the interval (-os, as) you would type: (-inf, info 1 8) For the IVP (+²36) dy and dy doe" de 3 x 436 dx +y=sin(u) Y(3)= 849, Y'(3)=13, y" (3)= 7, y"(3) = 10 an (21)= au (u) = x²36 2 Now au 1) = 0 => x-36=0 +x=16 These values of a divides the internal (500,00) into three parts (-+-6), (6, 6) and (6,00). Now initial conditions are specified at 2=3, which lies in the interval 66,6). Hence the fundamental existence therem for linear differential equations guarantees the existence of a unique solution on the interval (-6,6) Coefficients adu), a, (2), 934), au (1) and g(x) are continuous on (-6,6). as all the

solution: (1) for the interval (-os, as) you would type: (-inf, info 1 8) For the IVP (+²36) dy and dy doe" de 3 x 436 dx +y=sin(u) Y(3)= 849, Y'(3)=13, y" (3)= 7, y"(3) = 10 an (21)= au (u) = x²36 2 Now au 1) = 0 => x-36=0 +x=16 These values of a divides the internal (500,00) into three parts (-+-6), (6, 6) and (6,00). Now initial conditions are specified at 2=3, which lies in the interval 66,6). Hence the fundamental existence therem for linear differential equations guarantees the existence of a unique solution on the interval (-6,6) Coefficients adu), a, (2), 934), au (1) and g(x) are continuous on (-6,6). as all the